Fredholm, chaos-based and Gram-Charlier methods in fixed income derivative pricing

Abstract

This thesis deals with three issues of fixed income derivative pricing. Chapter 2 deals with bond pricing in mean-reverting CIR model, which is linked to quadratic functionals of Brownian motion. By the bivariate Laplace transform of quadratic functionals of the form (∫_0^T▒X_t dB_t,∫_0^T▒X_t^2 dt), where (X_t )_(t∈R_+ ) is an Ornstein-Uhlenbeck process driven by a standard Brownian motion (B_t )_(t∈R_+ ) and new bond pricing formulas are obtained as particular cases. Our method of computing the Laplace transform combines PDE arguments with Carleman-Fredholm determinant of associated Volterra operators that are computed by Fredholm expansions. In Chapter 3, we study bond pricing and spot forward rate models under the normal martingale setting, which has the chaotic representation property and satisfies the specified structure equation. We first extend the Wiener chaos-based framework to normal martingale chaos-based framework, then we derive the variance representation of price density V_t, which depends on the square-integrable random variable〖 X〗_∞. We obtain the spot forward rate chaos models by the chaos expansion of X_∞. Next we parameterize chaos coefficients in the spot forward rate models and do first chaos and second chaos model calibration. Chapter 4 deals with synthetic Collateralized Debt Obligation (CDO) pricing, which amounts to the computation of the expected tranche losses. We compute the expected γ-th tranche loss E[J_t^((γ) ) ] of CDOs with random recovery rates in Gram-Charlier expansion way and get the loss density function. In addition, we compare the density functions of the loss J_t approximated by Gram-Charlier expansion with Monte Carlo simulation.